1) y= 6(x 2 + 1) 3 u= x 2 + 1 y= 6u 3 dy/dx= y`= 6(3u 2 du/dx) y`= 18(x 2 +1) 2 d (x 2 +1) dx y`= 18(x 2 +1) 2 * 2x y`= 36x(x 2 +1) 2.

  • Published on
    24-Jan-2016

  • View
    218

  • Download
    0

Transcript

1) y= 6(x2 + 1)3 u= x2 + 1 y= 6u3 dy/dx= y`= 6(3u2 du/dx) y`= 18(x2+1)2 d (x2+1) dx y`= 18(x2+1)2 * 2x y`= 36x(x2+1)22) y= ln5(sen45x + x3) u= sen45x+x3 y= ln5u y`= 5ln4u d (ln u) dx y`= 5(ln4u) 1/u * du/dx y`= 5(ln4u) * 1/u * d (sen45x +x3) dx y`= 5(ln4u)*1/u [4(sen35x) * d (sen5x)+3x2] dx y`= 5 [ln4 (sen45x + x3)] * [4(sen35x)(cos5x) d (5x) + 3x2] sen45x + x3 dx y`= 5 [ln4 (sen45x + x3)] * [4(sen35x)(cos5x)5 + 3x2] sen45x + x3 3) y = 4x3-2x+1 Sea y= u u= 4x3-2x+1 x2+5 v v= x2 + 5 y`= v du/dx u dv/dx v2 (x2+5) d (4x3-2x+1) (4x3-2x+1) d (x2+5) dx dx y`= ---------------------------------------------------- . ( x2 + 5)2y`= (x2+5)(12x2-2) (4x3-2x+1)2x (x2+5)2y`= 12x4+58x2-10-(8x4-4x2+2x) (x2+5)2y`= 4x4+62x2-2x-10 (x2+5)2Otro mtodo: Y= u*v-1Y`= u d (v-1) + v-1 du dx dxY`= (4x3-2x+1) (-1)v-2 dv + v-1 d (4x3-2x+1) dx dxY`= (4x3-2x+1) (-2x) + 12x2 - 2 (x2+5)2 (x2+5) 4) y = 4e5x 6x 3xy= 1/3 [(4e5x 6x )x-1] y`= 1/3[(4e5x-6x) d (x-1) + x-1 d (4e5x-6x)] dx dxy`= 1 [ (4e5x- 6x) + 4 * 5e5x (ln6)6x ] 3 -x2 x DERIVACIN IMPLCITASea f(x, y) = 01) Se despeja y en funcin de x y se procede a efectuar la derivacin.2) Si el despeje de y en funcin de x resulta complicado, se procede a derivar en forma implcita. Ej: x2 + y2 4 = 0Caso 1) y2= 4 x2Y= (4 x2)1/2Y`= dy/dx= (4-x2)-1/2 * (-2x) dy/dx= -x/yCaso 2) y2= 4 x2 x2 + y2 - 4 = 0 dx2/dx + dy2/dx d4/dx=02x dx/dx + 2y dy/dx = 02y dy/dx = -2xdy/dx= -x/y 1) x3 y2 - 4xy4 - 3x2 y2 + senxy - 1= 0 x3 *2y dy/dx +3x2y2 -4x*4y3 *dy/dx- 4y4 - 3x2 *2ydy/dx -6xy2 +[cos(xy)] (xdy/dx + y)= 0dy = [2x3y - 16xy3-6x2y+xcos(xy)] = 4y4 - 3x2y2 + 6xy2-y cos xydxdy = 4y4-3x2y2+6xy2-ycosxydx 2x3 y-16xy3-6x2y+xcos(xy)3x(x2 - y2)1/2 + tg x/y - 3 =03x *1/2 (x2 - y2) -1/2 (2x-2y dy/dx) +3 (x2 - y2)1/2 + (sec2x/y) (y-x dy/dx) =0 y2 3x2 +3(x2-y2)1/2 + 1 sec2 x dy = (x2-y2)1/2 y y dx 3xy + x sec 2 x (x2- y2)1/2 y2 y

Recommended

View more >